A multiresolution analysis of the relationship between spatial distribution of reservoir parameters and time distribution of data measurements

An important issue in reservoir parameter estimation is to develop computationally efficient and reliable nonlinear regression procedures. We present a multiresolution wavelet approach to estimate spatial distribution of reservoir parameters, by performing the nonlinear least squares procedure in the wavelet domains of both time and space. Wavelet transforms have the ability to reveal important events in any signal or image. Thus we transformed both the model space and the data time into spatial wavelet and time wavelet domains and used a thresholding to select a subset of wavelet coefficients from each of the transformed domains. These subsets were subsequently used in the nonlinear regression procedure to estimate the appropriate subset of reservoir parameters. The appropriate subset is not only smaller and therefore more efficient computationally, the problem is also reduced to the consideration of only the important components of the measured data and only the part of the reservoir description that depends on them. We applied the procedure to a radially composite reservoir system to demonstrate the reliability of the approach. The inverse problem was solved to estimate the distributed permeability values by performing the nonlinear least square regression in the wavelet domains (time and space). Results obtained were compared to those obtained from the conventional nonlinear regression approach. The time-space wavelet approach proves to be more efficient computationally compared to the conventional approach. By reducing the dimensions of the model and data spaces the model stabilizes the algorithm and gives faster convergence. Significantly, the approach reveals the true number of reservoir parameters that can be appropriately estimated from a given data set. The approach provides a good means to integrate different data properly while avoiding the inclusion of irrelevant data during nonlinear regression. The procedure also reduces the number of iterations needed for convergence.